Hermitian form over a division ring
Let be a division ring admitting an involution (http://planetmath.org/Involution2) . Let be a vector space
over . A Hermitian form
over is a function from to , denoted by with the following properties, for any and :
- 1.
is additive in each of its arguments,
- 2.
,
- 3.
,
- 4.
.
Note that if the Hermitian form is non-trivial and if is the identity on , then is a field and is just a symmetric bilinear form.
If we replace the last condition by , then over is called a skew Hermitian form.
Remark. Every skew Hermitian form over a division ring induces a Hermitian form and vice versa.